Download On - Open Set in Ideal Topological Spaces

Journal of Progressive Research in Mathematics(JPRM)
ISSN: 2395-0218
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Volume 6, Issue 2
Published online: December 28, 2015|
Journal of Progressive Research in Mathematics
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On
- Open Set in Ideal Topological Spaces
A. E. Radwan 1, A. A. Abd- Elgawad 2, H. Z. Hassan3
1
Department of mathematics, Faculty of Science, Ain Shams University, Egypt.
Runing addres: Department of mathematics, Faculty of Science and Arts(UNAIZAH)
Alqassim University,Saudi Arabia.
2
Department of mathematics, Faculty of Science, Suze Canal University, Egypt. ‫‫‬
3Department of mathematics, Faculty of Science, Suze Canal University, Egypt.
Abstract
The aim of this paper is to study separation axioms and compactness of Iα- open set in ideal topological
spaces, which was introduced by M.E. Abd El-Monsef, etc [1].
Keywords:
and
- open set;
-
- space;
-
- space;
-
- space;
-
- space;
-
- space
- compact space.
1. Introduction
The subject of ideals in topological spaces has been studied by Kuratowski [3] and Vaidyanathaswamy [7]. An
ideal
on a set
is a nonempty collection of subsets of
which satisfies: (1)
and
implies
and (2)
,
implies
Given a topological space
with ideal
operator
and if
is the set of all subsets of . A set
with respect to
,
,
Kuratowski closure operator
for the topology
defined by
for
. If
is an ideal on
In an ideal topological space
closed subsets of
in
where
, called the
, if
then
.
-topology and finer than
then
For any ideal topological space
, is
for
is called an ideal topological space.
will denote the interior of
are called - closed sets. A subset
- closed if and only if
elements of
and , is defined as follows:
[6]. When there is no chance for confusion we will simply write
and
so
on
, called a local function [3] of
for
set if
.
in
of an ideal topological space
. The
is
[2].
, the collection
are called - open sets. A subset
is a basis for
of an ideal topological space
. It is clear that, in an ideal topological space
, if
[2]. The
is said to be - dense
then
and
.
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Journal of Progressive Research in Mathematics(JPRM)
ISSN: 2395-0218
Recall that, if
is an ideal topological space and
relative topology on
and
if
. A subset
. The family of all
called the
- topology on
space
is closed then
of a space
for
-
local
, where
is said to be
- open subsets of a space
and denoted by
is the
- open set
forms a topology on
,
. It is finer than . If every nowhere dense set in a
[5].
The concept of a set operator
an
then
, is an ideal topological subspace.
Given a topological space
called
is a subset of
was introduced by A. A. Nasef [4] in 1992 which is
function
of
with
respect
to
.
It
was
defined
as
follows:
,
where
. When there is no chance for confusion we will simply write
. An
- closure operator, denoted by
topology, finer than
which is called the -
. It is defined as follows:
ambiguity we will simply write
. A basis
. We will denote by
and closure of
with respect to
in
are called
- closed (respectively,
- closed sets. A subset
if
and
. The elements of
- dense) if and only if
topological space
-
. When there is no
for
follows:
subsets of
, for a topology
for
for
is described as
and
the interior
are called
- open sets. Closed
of an ideal topological space
(respectively,
then
is
). In an ideal
and
. So
.
Given a topological space
The family of all
. A subset
of
is said to be
- open subsets of a space
topological spaces
and
- open set if
is denoted by
has the same property
an ideal topological space
possesses property
[1]. Consider the ideal
and define a function
such that
irresolute‫‫‬homeomorphism,‫‫‬i.e…,‫‫‬then‫‫‬if‫‫‬the‫‫‬ideal topological space
topological space
.
is said to be
then
has any property
is called
is
and the ideal
- topological property. A property
- hereditary property if and only if every
-
- subspace of
of
also
.
In the following subjects we need to remember some concepts introduced by Radwan in 2015 [6]. An
-
boundary set is defined as: Let
-
boundary point of
be an ideal topological space and
if for every
is nonempty set. The set of all
denoted by
-
. Let
defined by the union of
- open neighborhood set for
- boundary points of
and
-derived set of
is
- open set in
the ideal topological spaces
function. Then if
.
is said to be
is said to be an
satisfies that the intersection with
is called
-boundary set of
be an ideal topological space and
be a function.
- open set in
.
. Then
and
and simply is
- closure set of
and simply is denoted by
-
is
. Let
- irresolute function if the inverse image of every
- topological property is a new property which defined as Consider
and
satisfies any property
such that
so did
then this property is called
is
- irresolute
- topological property.
2. SEPARATION AXIMOS IN Iα- OPEN SETS
In this section, we will study some properties of different types of separation axioms on the level of
ideal topological spaces using Iα- open sets.
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Journal of Progressive Research in Mathematics(JPRM)
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Definition 2.1 Let
elements in
element.
be an ideal topological space then it is called
there exist an
Proposition 2.2
- open set in
is an
-
-
- space if for any two different
containing one element of them but does not contain the other
- space if and only if
-
-
for any
such that
.
Remark 2.3 The
Example
-
- property is not hereditary property.
2.4
Consider
the
ideal
topological
then
take
, then
Corollary 2.5 The
-
is not
- property is
2.7
Consider
the
. It is clear that
- - space for
is
- - space but if we
-
- space is not necessary to be
ideal
topological
-
as the following example shows.
spaces
and
such
a
that
.
and
Define
that
.
and
Then
such
- hereditary property.
Remark 2.6 The continuous image of
Example
space
.
function
such
Then
,
that
.
and
. Thus
continuous function and it is clear that
the only
Remark 2.8 The
Example
2.9
-
is
- - space but
- open set contains
is
the
ideal
topological
- space for
but it is also contain
spaces
and
and
such
.
such that
but
is not because, if we take
also contain
.
Corollary 2.10 The
-
Remark 2.11 Every
example shows.
that
.
and
Define a function
- property is
- space is
-
. Then
such that
the only
is
- open set contains
is
- space
but it is
- topological property.
- space but the converse is not necessary to be true, as the following
Example 2.12 Take the ideal topological space
and
-
-
- property is not necessary to be topological property, as the following example shows.
Consider
Then
is not
is
such that
,
. Then
.
From
we get that the space
Definition 2.13 Let
elements in there exist two
of those elements.
is not
- space but from
we get that the space is
be an ideal topological space. Then it is called
- open sets in
such that each
-
-
- space.
- space if for any two different
- open set of them containing only one element
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Journal of Progressive Research in Mathematics(JPRM)
ISSN: 2395-0218
Proposition 2.14
closed sets in
is an
-
- space if and only if for any element
-
is an
-
are
-
- space if and only if the following statements are
.
ii.
.
iii.
-
.
iv.
-
.
v.
-
.
Corollary 2.16
Remark 2.17
-
Corollary 2.18
- property is
Example
-hereditary property.
- property is not topological property, as the following example shows.
-
- property is
Remark 2.19 Every
example shows:
- topological property.
- space is
2.20
-
- space but the converse is not necessary to be true. As the following
Consider
the
ideal
topological
and
of
we have that
.
Corollary 2.15 An ideal topological space
equivalent:
i.
in
is
space
, then the family of all
. It is clear that
- space since, if we fix
then for any element
where
is
there is no open set
-
- open subsets
- space which is not
containing
but not
contains .
Remark 2.21 Every
example shows.
-
Example 2.22 The
-
- open sets in
Remark 2.27
Corollary 2.28
-
- space but it is not
such that each
-
- space for
such that
- open set of them containing only one element of
be an ideal topological space. It is called
elements in there exist two disjoint
element of those elements.
-
Corollary 2.26
- space but the converse is not necessary to be true, as the following
- property is not hereditary property.
Definition 2.24 Let
Remark 2.25
-
in Example 2.12 is
there is no two
those elements.
Remark 2.23
- space is
- open sets in
-
such that each
- space if for any two different
- open set of them containing only one
- property is not hereditary property.
-
-
- property is
- hereditary property.
- property is not topological property.
-
- property is
Remark 2.29 Every
- space is
Remark 2.30 Every
-
Example 2.31 The
- topological property.
-
- space is
- space but the converse is not necessary to be true.
-
- space is
in Example 2.12 is
there is no two disjoint
- open sets in
-
-
- space but the converse is not necessary to be true.
- space but it is not
such that each
-
- space for
such that
- open set of them containing only one
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element of those elements. the ideal topological space in Example 2.21 is
any two
- open sets in
be an ideal topological space. It is called
- closed set in
-
- space for
- regular space if for every element in
does not contain the previous element then there exist two disjoint
that one of them containing the element and the other set containing the
space.
Definition 2.33 Let
element in
- space but not
, the intersection of them is not empty set.
Definition 2.32 Let
and
-
such
- closed set. It is denoted by
be an ideal topological space. It is called almost
and every closed set in
- open sets in
- -
- regular space if for every
does not contain the previous element then there exist two disjoint
- open
sets in
such that one of them containing the element and the other set containing the closed set. It is denoted by
almost-
- -space.
Proposition 2.34
such that
is an
- space if and only if for any element
then there exists another
Proposition 2.35
is an almost
such that
- open set
-
-
Corollary 2.37
- property is
-
Remark 2.38
-
Corollary 2.39
-
in
in
.
and every open set
satisfying that
-
in
.
- property are not hereditary properties.
-
- property are not topological property.
-
- space
-
- space but the converse is not necessary to be true.
-
-space if
is
-
-space and
-space.
space and almost
be an ideal topological space then it is called almost
-
Corollary 2.44 Both almost
-
Proof Both
-
Corollary 2.47
Proof Both
-
-
-
Remark 2.48 Every
true.
-space if
is
-
-
-
-space then
is
-
-space.
- property are not hereditary property.
-
- property are not hereditary properties.
- hereditary property.
-
-
- property are
- property and
-
- property is
- property and
-
- property and
- property and
- property, almost
is
- property and
- property is
Corollary 2.46 Both almost
-
-
- property, almost
Corollary 2.54
-
-space.
Proposition 2.43 If the ideal topological space
Proof
in
be an ideal topological space then it is called
Definition 2.42 Let
Proof
satisfying that
- open set
- topological property.
- space is almost
Definition 2.41 Let
and every
- hereditary property.
- property is
Remark 2.40 Every
- open set
- property and almost
- property and almost
-
in
in
- space if and only if for any element
then there exists another
Remark 2.36 Both
-
-
- property and
- hereditary properties.
-
- property are not topological properties.
-
- property are not topological properties.
- topological property.
-
- property are
- space is almost
-
- topological properties.
- space is
-
- space but the converse is not necessary to be
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Definition 2.49 Let
disjoint
be an ideal topological space then it is called
- closed sets in
one of the two disjoint
there exist two disjoint
- closed sets in
- open sets in
. It is denoted by
- normal space if for every two
such that each
- -space.
Definition 2.50 Let
be an ideal topological space then it is called almost
two disjoint closed sets in
there exist two disjoint
one of the two disjoint closed sets in
Proposition 2.51
in
is an
containing
containing
is almost
-
- open set
- property and almost
2.54
Consider
such that each
in
- closed set
-
the
in
- -
-
space
- space and almost
nor
almost
topological
space
- open set
-
.
and every open set
-
in
.
space
such
. Then
that
. It is clear that
then
is neither
for
such that
but
in
-space but if we take
- -
and every
- property are not hereditary properties.
ideal
-
in
satisfying that
and
is
- open set contain only
satisfying that
- space if and only if for any closed set
then there exists another
Remark 2.53 Both
Example
-
- open set
- normal space if for every
- -space.
- space if and only if for any
then there exists another
Proposition 2.52
- open sets in
. It is denoted by almost
-
- open set contain only
,
and
then
,
.
Corollary 2.55
-
- property is
Remark 2.56 Both
-
Corollary 2.57
- property is
-
Remark 2.58 Every
- hereditary property.
- property and almost
- property are not topological property.
- topological property.
- space is almost
Definition 2.59 Let
-
-
- space
-
- space but the converse is not necessary to be true.
be an ideal topological space. It is called
-
-space if
is
-
-space and
-
-space.
Definition 2.60 Let
and almost
-
be an ideal topological space. It is called almost
-
-space is
-
Proposition 2.62 Every
-
-space is
-
Remark 2.63 Both almost
-
-
-
-
- property, almost
Corollary 2.66
is
-
-space
-
Remark 2.67 Every
-space.
-space.
- property and
-
- property is
Remark 2.65 Both almost
Proof
-
- property, almost
Corollary 2.64
-space if
-space.
Proposition 2.61 Every
Proof
-
- property and
- property is
- property are not hereditary properties.
-
- property are not hereditary properties.
- hereditary property.
- property and
-
-
-
- property and
- property are not topological properties.
-
- property are not topological properties.
- topological property.
- space is almost
-
- space is
-
- space but the converse is not necessary to be
true.
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3.
- COMPACT SPACES
In this section we will study
- compact spaces using the concept
Definition 3.1 A subset
by
of an ideal topological space
- open sets in
for
. A space
Remark 3.2
- open sets.
is
is said to be
- compact if for every cover
, there is a finite subfamily
- compact if
is
of
such that
- compact as a set.
- compact property is not hereditary property.
Example
3.3
Consider
the
ideal
topological
space
such
and
. The ideal topological space
is
not
-
compact
space
is an
that
.
Then
- compact space and by taking
for:
is
open
which
cover
for
but
.
Remark 3.4
- compact property is neither topological property nor
Example
3.5
Consider
the
ideal
topological
and
spaces
and
. Then
Define a function
function and
- topological property.
- irresolute function and
is
is
that
.
such that
is also continuous.
such
thus
- compact space but
- irresolute function. Hence
is continuous, 1-1, onto
is not. We can see that
is also
- compact property is neither topological nor
- topological property.
Corollary 3.6 If
Remark 3.7 If
such that
are
- compact sets over
such that
are
- compact sets over
then
is also
- compact set.
then it is not necessary that
to be
- compact set. As the following example shows.
Example
3.8
Consider
the
ideal
topological
space
such
that
and
Then
. The ideal topological space
so
are
is an
.
- compact space and by taking
- compact sets over
but
which is not
-
compact space or
-
compact set.
Remark 3.9 There is no relation between being the ideal topological space
compact space.
Example 3.10
i.
Consider
the
ideal
topological
space
such
that
and
Then
. We know that
is not compact space but the ideal topological space
compact space.
ii.
Consider the ideal topological space
such that
Then
. We know that
is not
is not
- compact space but
is
and
.
is compact space but the ideal topological space
- compact space. Now, take
is finite subfamily of
.
is an
- open cover for
but
, then there exists
, and it is infinite. Hence
is compact.
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ISSN: 2395-0218
References
[1] M. E. Abd El-Monsef, A. E. Radwan and A. I. Nasir, Some generalized forms of compactness in ideal
topological spaces, Archives Des Sciences, Vol.66, no.3, Mar 2013,334-342.
[2] D. Jankovic, T. R. Hamlett, New topologies from old via ideals, Amer, Math, Monthly, 97(1990), 259-310.
[3] K. Kuratowski, Topology, Vol 1, New York: Academic Press, (1933).
[4] A. A. Nasef, Ideals in general topology, Ph.D, Thesis, Tanta University, (1992).
[5] O.Njastad, On some classes of nearly open sets, Pacific J, Math. 15(1965), 961-970.
[6] A. E. Radwan, A. A. Abd- Elgawad‫‫‬and‫‫‬H.‫‫‬Z.‫‫‬Hassan,‫‫‬On‫‫‬Iα- connected spaces via ideal, Journal of Advances
in Mathematics.(2015),3006-3014
[7] V.Vaidyanathaswamy, The localization theory in set topology, Pro. Indian Acad. Sci, 20(1945),51-61.
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